If a point z lies on line between two points x and y such that xz=yz. Then prove that xz=1/2 xy

To prove that \( xz = \frac{1}{2} xy \) given that \( xz = yz \), let's go through the following steps:

### 1. Represent Points on the Number Line
Let's assume \( x \), \( y \), and \( z \) are real numbers representing points on a number line.

### 2. Given Condition
The given condition is that \( xz = yz \).

This implies that the distance from \( x \) to \( z \) is equal to the distance from \( z \) to \( y \).

### 3. Expressing the Distances
The distance between two points \( a \) and \( b \) on a number line is given by \( |a - b| \).

So, the distances can be expressed as:
\[
xz = |x - z|
\]
\[
yz = |y - z|
\]

Given that \( xz = yz \), we have:
\[
|x - z| = |y - z|
\]

### 4. Solving the Equation
The equation \( |x - z| = |y - z| \) implies two possible cases:
1. \( x - z = y - z \)
2. \( x - z = -(y - z) \)

#### Case 1: \( x - z = y - z \)
If \( x - z = y - z \), subtract \( z \) from both sides:
\[
x = y
\]
This case is trivial and does not provide any meaningful result for distinct points.

#### Case 2: \( x - z = -(y - z) \)
If \( x - z = -(y - z) \), simplify:
\[
x - z = -y + z
\]

Add \( y \) to both sides:
\[
x + y = 2z
\]

Now, solve for \( z \):
\[
z = \frac{x + y}{2}
\]

### 5. Express \( xz \) and \( xy \)
Now that we have \( z = \frac{x + y}{2} \), we can express \( xz \) and \( xy \) as follows:
\[
xz = |x - z| = \left|x - \frac{x + y}{2}\right| = \left|\frac{2x - x - y}{2}\right| = \left|\frac{x - y}{2}\right| = \frac{|x - y|}{2}
\]
\[
xy = |x - y|
\]

### 6. Conclusion
Since \( xz = \frac{|x - y|}{2} \) and \( xy = |x - y| \), we have:
\[
xz = \frac{1}{2} xy
\]

Thus, the statement \( xz = \frac{1}{2} xy \) is proven.

---

Would you like further details on any step, or do you have any questions?

Here are some related questions:

1. How does the concept of the midpoint relate to the result \( z = \frac{x + y}{2} \)?
2. What are other conditions that can be derived if \( xz = k \times xy \) where \( k \) is a constant?
3. How can this proof be extended to points in a plane rather than just on a number line?
4. What happens if \( x \) and \( y \) are complex numbers instead of real numbers?
5. How would this situation be represented on a coordinate plane?

**Tip:** Always consider the geometric interpretation of algebraic results, as it often provides deeper insight into the problem.

If a point z lies on the line between two points x and y such that xz = yz, then prove that xz = 1/2 xy.

This problem explores the concept of equidistant points on a number line. Learn how to prove that xz = 1/2 xy when z lies on the line between x and y, and xz equals yz. Suitable for students in Grades 9-11, this problem emphasizes the midpoint formula and the geometric interpretation of distance.

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